$(40$ points$)$

$2.$ An air$-$conditioning system is to take in outdoor air $($state $1)$ at $10C,30$ percent relative humidity and a humidity ratio of $0\mathrm{.}0023.$ It is conditioned to $($state $3)$ humidity ratio of $0\mathrm{.}0121.$ The steady intake volumetric flow rate of dry air is $45\frac{m^3}{m}in.$ The outdoor air is first heated to $($state $2)20C$ in the heating section and then humidified by the injection of hot water mist in the humidifying section. Assuming the entire process takes place at a pressure of $100$ kPa $,$ determine the following $($do not use the Psychometric chart$)$:

$C_{p,\text{a}\text{i}\text{r}}=1\mathrm{.}005\frac{kJ}{k}gK,R_{\text{a}\text{i}\text{r}}=0\mathrm{.}287\frac{kJ}{k}gK$

$55\mathrm{.}375k\frac{g}{m}in($a$)$ mass flow rate of dry air $(15)$

$6\mathrm{.}617$ kW

$($b$)$ the required heat transfer rate in the heating section $(15)$

$g\frac{g}{m}in($c$)$ the required mass flow rate of hot steam in the humidifying section $(10)$